- \( \displaystyle i^{1937} + i^{1938} + i^{1939} = \)
Solution
\( \displaystyle -1\)
- \( \displaystyle 4\sqrt{-\frac{7}{12}} - 3 \sqrt{-\frac{1}{24}} \)
Solution
\( \displaystyle -\frac{i\sqrt{6}}{4} + \frac{2i\sqrt{21}}{3} \)
- \( \displaystyle \left( 3 - i \right) ^{-2} \)
Solution
\( \displaystyle \frac{2}{25} + \frac{3}{50}i \)
- Given \( \displaystyle a = -\frac{5}{6} + \frac{1}{2}i\) and \( \displaystyle b = \frac{2}{3} + \frac{3}{4}i,\) find \(\overline{a} + b\)
Solution
\( \displaystyle -\frac{1}{6} + \frac{1}{4}i \)
- Factor \( \displaystyle x^4 + 5x^2 + 6 \) over the complex numbers
Solution
\( \displaystyle \left( x + i\sqrt{3} \right) \left( x - i\sqrt{3} \right) \left( x + i\sqrt{2}\right) \left( x - i\sqrt{2}\right) \)
- Given \( \displaystyle z = -4 + 3i, \) express \( \displaystyle z^{-1} \) as a complex number
Solution
\( \displaystyle -\frac{4}{25} - \frac{3}{25}i\)
- Solve for \(x\) and \(y:\) \( \displaystyle \frac{1}{2}x - 5y - \left(7y - \frac{3}{2}x\right)i = 32 + 45i \)
Solution
\( \displaystyle x = \frac{1}{4}, y=-\frac{51}{8} \)
- Solve for real values of \(x\) and \(y:\) \( \displaystyle 3y + yi = 2i - xi + x \)
Solution
\( \displaystyle x = \frac{3}{2}; y = \frac{1}{2} \)
- Find the sum and difference of \( \displaystyle 7 + 5i \) and its conjugate.
Solution
Sum: 14; Difference: \( 10i \)
- Express each of the following as a complex number given \( \displaystyle a = 5 - 8i; b = -7 + 3i; c = 2 + 8i; d = -4i \)
- \( a + b\)
Solution
\( \displaystyle -2 - 5i \)
- \( a + c \)
Solution
7
- \( a - c \)
Solution
\( \displaystyle 3 - 16i \)
- \( c - d \)
Solution
\( \displaystyle 2 + 12i \)
- \( \overline{c} - d \)
Solution
\( \displaystyle 2 - 4i \)
- \( d - \overline{a}\)
Solution
\( \displaystyle -5 - 12i \)
- Factor completely over the complex numbers: \( \displaystyle a^2 + 4 \)
Solution
\( \displaystyle (a + 2i)(a - 2i) \)
- Factor completely over the complex numbers: \( \displaystyle x^4 + 7x^2 + 12 \)
Solution
\( \displaystyle \left(x + 2i\right)\left(x - 2i\right)\left(x + i\sqrt{3}\right)\left(x - i\sqrt{3}\right) \)
- Express the reciprocal of \( \displaystyle 2 - 5i \) as a complex number.
Solution
\( \displaystyle \frac{2}{29} + \frac{5}{29}i \)
- Express \( \displaystyle \frac{2 + 3i}{7 + 4i} \) as a complex number.
Solution
\( \displaystyle \frac{2}{5} + \frac{1}{5}i \)
- \( \displaystyle (4 - 7i)(3 + i) = \)
Solution
\( \displaystyle 19 - 17i \)
- \( \displaystyle \left(3 - 4i \right)^2 = \)
Solution
\( \displaystyle -7 - 24i \)
- \( \displaystyle \left(-1 + i\sqrt{5}\right)^2 = \)
Solution
\( \displaystyle -4 - 2i\sqrt{5} \)
- \( \displaystyle \frac{3 + i}{4 - i} \)
Solution
\( \displaystyle \frac{11}{17} + \frac{7}{17}i \)
- \( \displaystyle \frac{8i}{1 + 3i} \)
Solution
\( \displaystyle \frac{12}{5} + \frac{4}{5}i \)
- \( \displaystyle \frac{3 - 6i}{-2 - 5i} \)
Solution
\( \displaystyle \frac{24}{29} + \frac{27}{29} i\)